Finite Group Databasebeta

← C2×#11727256 #13353580

Subgroups — C2×#11727256 #13353580

Order 1792.

1878 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic7C211yesmaximal, characteristic
2C2×C7×#380C211yesmaximal, characteristic
3C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic
4C2×#89324C214yesmaximal, complemented
5#11727256C218yesmaximal, direct product factor
6C24×C4×C7C2211yescharacteristic
7C24×Dic7—21no—
8C22×C7×C4⋊C4C2212yes—
9C22×C7×C22⋊C4—21no—
10C22×C4⋊Dic7—24no—
11C22×C4⋊Dic7—21no—
12C22×C4⋊Dic7—22no—
13C22×C4⋊Dic7C418yescomplemented
14C22×C4⋊Dic7C2212yes—
15C22×C4⋊Dic7C2214yes—
16C23×C7⋊C8C2212yes—
17C7×#380—22no—
18C7×#380C2214yes—
19C2×C7×#95—21no—
20C2×C22⋊(C7⋊C8)—24no—
21C2×C22⋊(C7⋊C8)C2214yes—
22#89324C22116yescomplemented
23C2×#1824—71nomaximal, Sylow
24C25×C7D411yescharacteristic
25C23×C4×C7—28no—
26C23×C4×C7—22no—
27C23×C4×C7C2×C414yes—
28C23×C4×C7D414yes—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D412yes—
31C23×C4×C7C2312yes—
32C23×C4×C7C2311yescharacteristic
33C23×Dic7—28no—
34C23×Dic7—24no—
35C23×Dic7—22no—
36C23×Dic7—48no—
37C2×C7×C4⋊C4—28no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4C2314yes—
40C2×C7×C22⋊C4—24no—
41C2×C7×C22⋊C4—44no—
42C2×C4⋊Dic7—216no—
43C2×C4⋊Dic7—28no—
44C2×C4⋊Dic7—22no—
45C2×C4⋊Dic7—24no—
46C2×C4⋊Dic7—24no—
47C2×C4⋊Dic7—28no—
48C2×C4⋊Dic7C2×C418yes—
49C2×C4⋊Dic7C2×C4116yescomplemented
50C2×C4⋊Dic7C2314yes—
51C22×C7⋊C8—24no—
52C22×C7⋊C8—28no—
53C22×C7⋊C8C2314yes—
54C7×#95—24no—
55C22⋊(C7⋊C8)—28no—
56C23×C4⋊C4—71no—
57C22×C22⋊C8—71no—
58C2×#380D711yescharacteristic
59#1824—78no—
60C2×#340—74no—
61C24×C7—24no—
62C24×C7—24no—
63C24×C7C22⋊C414yes—
64C24×C7SD1614yes—
65C24×C7Q1614yes—
66C24×C7C2×D412yes—
67C24×C7C2×D411yescharacteristic
68C22×C4×C7—24no—
69C22×C4×C7—216no—
70C22×C4×C7—28no—
71C22×C4×C7—216no—
72C22×C4×C7—28no—
73C22×C4×C7—24no—
74C22×C4×C7—44no—
75C22×C4×C7—44no—
76C22×C4×C7—44no—
77C22×C4×C7C22⋊C414yes—
78C22×C4×C7C22⋊C418yes—
79C22×C4×C7C22×C414yes—
80C22×C4×C7C22×C412yes—
81C22×C4×C7C2×D412yes—
82C22×C4×C7C2×D414yes—
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D412yes—
85C22×C4×C7#4118yes—
86C22×C4×C7C2411yescharacteristic
87C22×Dic7—28no—
88C22×Dic7—216no—
89C22×Dic7—24no—
90C22×Dic7—416no—
91C22×Dic7—432no—
92C22×Dic7—48no—
93C7×C4⋊C4—28no—
94C7×C4⋊C4—44no—
95C7×C22⋊C4—48no—
96C4⋊Dic7—28no—
97C4⋊Dic7—216no—
98C4⋊Dic7—24no—
99C4⋊Dic7C22×C418yes—
100C2×C7⋊C8—24no—
101C2×C7⋊C8—28no—
102C2×C7⋊C8—48no—
103C24×C4—141no—
104C24×C4C2×D711yescharacteristic
105C23×C8—72no—
106C22×C4⋊C4—78no—
107C22×C4⋊C4—74no—
108C22×C4⋊C4—72no—
109C22×C4⋊C4—144no—
110C22×C4⋊C4—142no—
111C22×C4⋊C4—141no—
112C22×C4⋊C4C2×D712yes—
113C22×C22⋊C4—21no—
114C2×C22⋊C8—74no—
115C2×C22⋊C8—144no—
116#380—22no—
117#380C2×D714yes—
118C2×#95—21no—
119#340—716no—
120C23×C7—24no—
121C23×C7—24no—
122C23×C7—28no—
123C23×C7—28no—
124C23×C7—216no—
125C23×C7—216no—
126C23×C7#103116yes—
127C23×C7C2×C22⋊C414yes—
128C23×C7C2×C22⋊C412yes—
129C23×C7#12314yes—
130C23×C7C2×SD1614yes—
131C23×C7C2×SD1612yes—
132C23×C7C2×Q1614yes—
133C23×C7C2×Q1612yes—
134C23×C7AGL(1,ℤ/8)14yes—
135C23×C7C22×D411yescharacteristic
136C2×C4×C7—24no—
137C2×C4×C7—28no—
138C2×C4×C7—28no—
139C2×C4×C7—28no—
140C2×C4×C7—216no—
141C2×C4×C7—44no—
142C2×C4×C7—44no—
143C2×C4×C7—44no—
144C2×C4×C7—88no—
145C2×C4×C7C2×C22⋊C414yes—
146C2×C4×C7C2×C22⋊C412yes—
147C2×C4×C7#11718yes—
148C2×C4×C7#12318yes—
149C2×C4×C7#12314yes—
150C2×C4×C7C23×C411yescharacteristic
151C2×C4×C7C22×D411yescharacteristic
152C2×C4×C7C2×#4114yes—
153C2×Dic7—28no—
154C2×Dic7—432no—
155C2×Dic7—48no—
156C2×Dic7—416no—
157C7⋊C8—44no—
158C25C22⋊D711yescharacteristic
159C23×C4—28no—
160C23×C4—22no—
161C23×C4—148no—
162C23×C4—144no—
163C23×C4—142no—
164C23×C4—288no—
165C23×C4C4×D714yes—
166C23×C4D2814yes—
167C23×C4C22⋊D712yes—
168C23×C4C22⋊D711yescharacteristic
169C23×C4C22×D712yes—
170C23×C4C22×D711yescharacteristic
171C22×C8—74no—
172C22×C8—144no—
173C22×C8—148no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—78no—
177C2×C4⋊C4—716no—
178C2×C4⋊C4—74no—
179C2×C4⋊C4—148no—
180C2×C4⋊C4—1416no—
181C2×C4⋊C4—148no—
182C2×C4⋊C4—144no—
183C2×C4⋊C4—144no—
184C2×C4⋊C4—142no—
185C2×C4⋊C4C22×D714yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—44no—
188C22⋊C8—148no—
189#95—24no—
190C22×C7—28no—
191C22×C7—28no—
192C22×C7—232no—
193C22×C7—24no—
194C22×C7—24no—
195C22×C7#38114yes—
196C22×C7C2×#10318yes—
197C22×C7C2×#10314yes—
198C22×C7#41814yes—
199C22×C7#48118yes—
200C22×C7#48418yes—
201C22×C7C22×C22⋊C411yescharacteristic
202C22×C7C2×#12312yes—
203C22×C7C22×SD1611yescharacteristic
204C22×C7C22×Q1611yescharacteristic
205C22×C7C2×AGL(1,ℤ/8)12yes—
206C4×C7—24no—
207C4×C7—84no—
208C4×C7#38114yes—
209C4×C7C2×#11712yes—
210C4×C7C2×#12312yes—
211Dic7—48no—
212C24—24no—
213C24—24no—
214C24(C2×D7)⋊C414yes—
215C24#114214yes—
216C24#114414yes—
217C24C2×C22⋊D712yes—
218C24C2×C22⋊D711yescharacteristic
219C22×C4—24no—
220C22×C4—216no—
221C22×C4—216no—
222C22×C4—28no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—1416no—
229C22×C4—148no—
230C22×C4—144no—
231C22×C4—2816no—
232C22×C4—2832no—
233C22×C4—288no—
234C22×C4(C2×D7)⋊C414yes—
235C22×C4(C2×D7)⋊C418yes—
236C22×C4C2×C4×D714yes—
237C22×C4C2×C4×D712yes—
238C22×C4C2×D2814yes—
239C22×C4C2×D2812yes—
240C22×C4#115918yes—
241C22×C4C2×C22⋊D712yes—
242C22×C4C2×C22⋊D711yescharacteristic
243C22×C4C23×D711yescharacteristic
244C2×C8—144no—
245C2×C8—148no—
246C2×C8—288no—
247C4⋊C4—28no—
248C4⋊C4—44no—
249C4⋊C4—78no—
250C4⋊C4—1416no—
251C4⋊C4—148no—
252C4⋊C4—144no—
253C22⋊C4—48no—
254C2×C7—28no—
255C2×C7#182418yescomplemented
256C2×C7C2×#38111yescharacteristic
257C2×C7C22×#10311yescharacteristic
258C2×C7C2×#41811yescharacteristic
259C2×C7C2×#48112yes—
260C2×C7C2×#48412yes—
261C23—28no—
262C23—24no—
263C23—24no—
264C23—28no—
265C23—216no—
266C23—216no—
267C23#6384116yes—
268C23C2×(C2×D7)⋊C414yes—
269C23C2×(C2×D7)⋊C412yes—
270C23C2×#114214yes—
271C23C2×#114212yes—
272C23#649814yes—
273C23C2×#114414yes—
274C23C2×#114412yes—
275C23#651214yes—
276C23C22×C22⋊D711yescharacteristic
277C2×C4—28no—
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—216no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—88no—
286C2×C4—148no—
287C2×C4—2832no—
288C2×C4—288no—
289C2×C4—2816no—
290C2×C4#644018yes—
291C2×C4#644918yes—
292C2×C4C2×(C2×D7)⋊C414yes—
293C2×C4C2×(C2×D7)⋊C412yes—
294C2×C4#649814yes—
295C2×C4C22×C4×D711yescharacteristic
296C2×C4C22×D2811yescharacteristic
297C2×C4C2×#115914yes—
298C8—284no—
299C7C2×#182411yescharacteristic, complemented, Sylow
300C22—24no—
301C22—28no—
302C22—24no—
303C22—232no—
304C22—28no—
305C22#8971514yes—
306C22C2×#638418yes—
307C22C2×#638414yes—
308C22#8977314yes—
309C22#8980618yes—
310C22#8981618yes—
311C22C22×(C2×D7)⋊C411yescharacteristic
312C22C22×#114211yescharacteristic
313C22C2×#649812yes—
314C22C22×#114411yescharacteristic
315C22C2×#651212yes—
316C4—24no—
317C4—84no—
318C4—288no—
319C4#8971514yes—
320C4C2×#644012yes—
321C4C2×#644912yes—
322C2—28no—
323C2#1172725618yesdirect product factor
324C2C2×#8971511yescharacteristic
325C2C22×#638411yescharacteristic
326C2C2×#8977311yescharacteristic
327C2C2×#8980612yes—
328C2C2×#8981612yes—
subgroupG/H
centerC24C2×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#380C2
socleC24×C7C2×D4
prime radicalC25×C7D4
cosocleC22×C4×C7C24
hypercenterC2×#380D7
hypocenterC7C2×#1824
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#89324C2semidirect
5. #11727256C2direct (N × H)
13. C22×C4⋊Dic7C4semidirect
22. #89324C22semidirect
49. C2×C4⋊Dic7C2×C4semidirect
255. C2×C7#1824semidirect
299. C7C2×#1824semidirect
323. C2#11727256direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C4⋊Dic72
C2×C7×#3802
C22×C22⋊(C7⋊C8)2
C2×#893242
#117272562
C2×#18247